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A TimeSpace Tradeoff for Sorting on NonOblivious Machines
, 1981
"... This paper adopts the latter strategy in order to pursue the complexity of sorting ..."
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Cited by 24 (2 self)
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This paper adopts the latter strategy in order to pursue the complexity of sorting
Size Space tradeoffs for Resolution
, 2002
"... We investigate tradeoffs of various important complexity measures such as size, space and width. We show examples of CNF formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results, the first of their ..."
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Cited by 22 (3 self)
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We investigate tradeoffs of various important complexity measures such as size, space and width. We show examples of CNF formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results, the first of their kind, have implications on the efficiency (or rather, inefficiency) of some commonly used SAT solving heuristics. Our proof
Narrow proofs may be spacious: Separating space and width in resolution (Extended Abstract)
 REVISION 02, ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY (ECCC
, 2005
"... The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously ..."
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Cited by 16 (7 self)
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The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously been studied and related to the resolution refutation size of unsatisfiable CNF formulas. Also, the refutation space of a formula has been proven to be at least as large as the refutation width, but it has been open whether space can be separated from width or the two measures coincide asymptotically. We prove that there is a family of kCNF formulas for which the refutation width in resolution is constant but the refutation space is nonconstant, thus solving a problem mentioned in several previous papers.
Performance of VLSI engines for lattice computations
 Complex Systems
, 1987
"... Abstract. We address t he problem of designin g an d building efficient custo m Vl.Slbesed processors to do computat ions on large multidimensional lat tices. The design t ra deoffs for two architectures which provid e practical engines for lattice updates are deri ved and an alyzed. We find t hat ..."
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Abstract. We address t he problem of designin g an d building efficient custo m Vl.Slbesed processors to do computat ions on large multidimensional lat tices. The design t ra deoffs for two architectures which provid e practical engines for lattice updates are deri ved and an alyzed. We find t hat I/O constit utes t he principal bottleneck of processors des igned for latt ice computations, and we derive upp er bounds on t hroughp ut for lattice updates based on Hong and Kung's graphpebbling argument t hat models I / O. In particular, we show that R = O(BS1 / d) , where R is the site update rate, B is t he main memory bandwidth, S is t he processor sto rage, and d is t he dimension of th e lattice. 1.
TIMESPACE TRADEOFFS IN NONBLOCKING PARALLEL COMMUNICATIONS †
"... In this paper, we examine the complexity of nonblocking switching networks with an emphasis on the relations between their costs and path– lengths. Much of the earlier work on the complexity of nonblocking networks deals with deriving bounds on the number of crosspoints in such networks. While such ..."
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In this paper, we examine the complexity of nonblocking switching networks with an emphasis on the relations between their costs and path– lengths. Much of the earlier work on the complexity of nonblocking networks deals with deriving bounds on the number of crosspoints in such networks. While such complexity bounds provide valuable information about nonblocking networks, the cost alone is not sufficient to fully characterize their performance. To better evaluate the performance of these networks this paper investigates the basic relations between their costs and path–lengths. It is shown that the cost, Cn, of an ninput nonblocking network for a given path–length, Sn, satisfies the inequality Cn ≥ Snn 1+1/Sn. It is also shown that, for a given cost, Cn, and fanout, f, the path–length of a nonblocking network satisfies the inequality Sn ≥ n log 2 f n/(fCn). It is further established that there exist rearrangeable Clos networks which are near optimal in the sense of these bounds. Finally, it is shown that no nonblocking network which is derived from a 3stage Clos network construction is optimal when compared to these bounds.
A TIMESPACE TRADEOFF FOR SORTING ON NONOBLIVIOUS MACHINES*
"... A model of computation is introduced which permits the analysis of both the time and space requirements of nonoblivious programs. Using this model, it is demonstrated that any algorithm for sorting n inputs which is based on comparisons of individual inputs requires timespace product proportion ..."
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A model of computation is introduced which permits the analysis of both the time and space requirements of nonoblivious programs. Using this model, it is demonstrated that any algorithm for sorting n inputs which is based on comparisons of individual inputs requires timespace product proportional to n2. Uniform and nonuniform sorting algorithms are presented which show that this lower bound is nearly tight. 1. Hotivation and Contraposition to Previous Research The traditional approach to studying the complexity of a problem has been to examine the amount of some single resource (usually time or space) required to perform the computation. In an effort to better understand the complexity of certain problems, recent attention has been focused on examining the trade off between the required time and space. This paper adopts the latter strategy in order to pursue the complexity of sorting. The vast majority of timespace tradeoffs recently demonstrated have been for "straightline " (or "oblivious") programs 1, 5, 7, 10, 11,13, 14, 15, that is, programs in which the sequence of operations is independent of the actual values of the inputs. In this model, "time " refers to the number of operations performed, and "space " to the number of auxiliary (i.e., noninput and nonoutput) registers used to store intermediate results. (To distinguish this usage of space from others which follow, this will be referred to as "data space".) The problem of sorting has been considered in this rontext by Tompa 15, who demonstrated that any oblivious algorithm which sorts n inputs requires timespace product ~(n2). Although oblivious sorting algorithms have been studied extensively (see Knuth 6, where they are called "sorting networks"), most sorting algorithms are nonoblivious; that is, they continually test and branch based on comparisons of input
NorthHolland SEARCHING AND PEBBLING
, 1983
"... Abstract. We relate the search number of an undirected graph G with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting (7. Towards this end, we introduce nodesearching, a slight variant of searching, in which an edge is cleared by placing ..."
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Abstract. We relate the search number of an undirected graph G with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting (7. Towards this end, we introduce nodesearching, a slight variant of searching, in which an edge is cleared by placing searchers on both of its endpoints. We also show that the minimum number of searchers necessary to nodesearch a graph equals its vertex separator plus one. Key words. Searching a graph, pebble games, layout parameters of a graph, vertex separator of a graph. 1.
Electronic Colloquium on Computational Complexity, Report No. 125 (2010) Understanding Space in Proof Complexity: Separations and Tradeoffs via Substitutions
, 2010
"... For current stateoftheart satisfiability algorithms based on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. In the field of proof complexity, these resources correspond to the length and space of resolution proofs for formulas in conjunct ..."
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For current stateoftheart satisfiability algorithms based on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. In the field of proof complexity, these resources correspond to the length and space of resolution proofs for formulas in conjunctive normal form (CNF). There has been a long line of research investigating these proof complexity measures, but while strong results have been established for length, our understanding of space and how it relates to length has remained quite poor. In particular, the question whether resolution proofs can be optimized for length and space simultaneously, or whether there are tradeoffs between these two measures, has remained essentially open apart from a few results in restricted settings. In this paper, we remedy this situation by proving a host of lengthspace tradeoff results for resolution in a completely general setting. Our collection of tradeoffs cover almost the whole range of values for the space complexity of formulas, and most of the tradeoffs are superpolynomial or even exponential and essentially tight. Using similar techniques, we show that these tradeoffs in fact extend (albeit with worse parameters) to the exponentially stronger kDNF resolution proof systems, which operate with formulas in disjunctive normal form with terms of bounded arity k. We also answer the open question
Rig: A simple, secure and flexible design for Password Hashing
"... Abstract. Password Hashing, a technique commonly implemented by a server to protect passwords of clients, by performing a oneway transformation on the password, turning it into another string called the hashed password. In this paper, we introduce a secure password hashing framework Rig which is ba ..."
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Abstract. Password Hashing, a technique commonly implemented by a server to protect passwords of clients, by performing a oneway transformation on the password, turning it into another string called the hashed password. In this paper, we introduce a secure password hashing framework Rig which is based on secure cryptographic hash functions. It provides the flexibility to choose different functions for different phases of the construction. The design of the scheme is very simple to implement in software and is flexible as the memory parameter is independent of time parameter (no actual time and memory tradeoff) and is strictly sequential (difficult to parallelize) with comparatively huge memory consumption that provides strong resistance against attackers using multiple processing units. It supports clientindependent updates, i.e., the server can increase the security parameters by updating the existing password hashes without knowing the password. Rig can also support the server relief protocol where the client bears the maximum effort to compute the password hash, while there is minimal effort at the server side. We analyze Rig and show that our proposal provides an exponential time complexity against the lowmemory attack.